Abnormal sub-riemannian geodesics : Morse index and rigidity
A. A. Agrachev; A. V. Sarychev
Annales de l'I.H.P. Analyse non linéaire (1996)
- Volume: 13, Issue: 6, page 635-690
- ISSN: 0294-1449
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topAgrachev, A. A., and Sarychev, A. V.. "Abnormal sub-riemannian geodesics : Morse index and rigidity." Annales de l'I.H.P. Analyse non linéaire 13.6 (1996): 635-690. <http://eudml.org/doc/78396>.
@article{Agrachev1996,
author = {Agrachev, A. A., Sarychev, A. V.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {abnormal extremum; variational problem; sub-Riemannian geodesics},
language = {eng},
number = {6},
pages = {635-690},
publisher = {Gauthier-Villars},
title = {Abnormal sub-riemannian geodesics : Morse index and rigidity},
url = {http://eudml.org/doc/78396},
volume = {13},
year = {1996},
}
TY - JOUR
AU - Agrachev, A. A.
AU - Sarychev, A. V.
TI - Abnormal sub-riemannian geodesics : Morse index and rigidity
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1996
PB - Gauthier-Villars
VL - 13
IS - 6
SP - 635
EP - 690
LA - eng
KW - abnormal extremum; variational problem; sub-Riemannian geodesics
UR - http://eudml.org/doc/78396
ER -
References
top- [1] A.A. Agrachev, Quadratic Mappings in Geometric Control Theory, in: Itogi Nauki i Tekhniki; Problemy Geometrii, VINITI, Acad. Nauk SSSR, Moscow, Vol. 20, pp. 11-205, 1988. English transl. in J. Soviet Math., Vol. 51, 1990, pp. 2667-2734. MR966201
- [2] A.A. Agrachev, Topology of Quadratic Mappings and Hessians of Smooth Mappings, in: Itogi Nauki i Tekhniki; Algebra, Topologia, Geometria; VINITI, Acad. Nauk SSSR, Vol. 26, 1988, pp. 85-124. Zbl0719.58006MR978394
- [3] A.A. Agrachev, The Second-Order Optimality Condition in the General Nonlinear Case, Matem Sbornik, Vol. 102, 1977, pp. 551-568. English transl. in: Math. USSR Sbornik, Vol. 31, 1977. Zbl0362.49016MR451084
- [4] A.A. Agrachev and R.V. Gamkrelidze, Second-Order Optimality Condition for the Time-Optimal Problem, Matem. Sbornik, Vol. 100, 1976, pp. 610-643. English transl. in: Math. USSR Sbornik, Vol. 29, 1976, pp. 547-576. Zbl0341.49007MR425719
- [5] A.A. Agrachev and R.V. Gamkrelidze, Exponential Representation of Flows and Chronological Calculus, Matem. Sbornik, Vol. 107, 1978, pp. 467-532. English transl. in: Math. USSR Sbornik, Vol. 35, 1979, pp. 727-785. Zbl0408.34044MR524203
- [6] A.A. Agrachev, R.V. Gamkrelidze and A.V. Sarychev, Local Invariants of Smooth Control Systems, Acta Applicandae Mathematicae, Vol. 14, 1989, pp. 191-237. Zbl0681.49018MR995286
- [7] A.A. Agrachev and A.V. Sarychev, On Abnormal Extremals for Lagrange Variational Problems, to appear in J. Mathematical Systems, Estimation and Control. Zbl0826.49012
- [8] A.A. Agrachev and A.V. Sarychev, On Abnormal Extremals for Lagrange Variational Problems (summary), J. Mathematical Systems, Estimation and Control, Vol. 5, 1995, pp. 127-130. Zbl0826.49012
- [9] V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York-Berlin, 1978. Zbl0386.70001MR690288
- [10] V.I. Arnol'd, A.N. Varchenko and S.M. Gusein-Zade, Singularities of Differentiable Mappings, Vol. 1, Birkhäuser, Boston, 1985. MR777682
- [11] B. Bonnard and I. Kupka, Théorie de singularités de l'application entree/sortie et optimalité des trajectoires singulières dans le problème du temps minimal, Preprint, 1990.
- [12] V. Gershkovich, Engel Structures on Four Dimensional Manifolds, University of Melbourne, Department of Mathematics, Preprint Series No. 10, 1992.
- [13] B.S. Goh, Necessary Conditions for Singular Extremals Involving Multiple Control Variables, SIAM J. Control, Vol. 4, 1966, pp. 716-731. Zbl0161.29004MR205719
- [14] V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, Rhode Island, 1977. Zbl0364.53011MR516965
- [15] G.W. Haynes and H. Hermes, Nonlinear Controllability via Lie Theory, SIAM J. on Control, Vol. 8, 1970, pp. 450-460. Zbl0229.93012MR277859
- [16] H.J. Kelley, R. Kopp and H.G. Moyer, Singular Extremals, in: G. Leitman Ed., Topics in Optimization, Academic Press, New York, N. Y., 1967, pp. 63-101. MR215153
- [17] A.J. Krener, The High-Order Maximum Principle and its Applications to Singular Extremals, SIAM J. on Control and Optimiz., Vol. 15, 1977, pp. 256-293. Zbl0354.49008MR433288
- [18] G. Lion and M. Vergne, The Weyl Representation, Maslov Index and Theta-series, Birkhäuser, Boston, 1980. Zbl0444.22005MR573448
- [19] R. Montgomery, Geodesics, Which Do Not Satisfy Geodesic Equations, Preprint, 1991.
- [20] M. Morse, The Calculus of variations in the large, Amer. Math. Soc., New York, NY, 1934. Zbl0011.02802JFM60.0450.01
- [21] A.V. Sarychev, Integral representation for the trajectories of control system with generalized right-hand side (in Russian), Differentsialnye Uravnenia, Vol. 24, 1988, pp. 1551-1564. English transl.: Differential Equations, Vol. 24, 1988, pp. 1021-1031. Zbl0674.34064MR965603
- [22] A.V. Sarychev, The index of the second variation of the control system, Matem. Sbornik, Vol. 113, 1980, pp. 464-486. English transl. in: Math. USSR Sbornik, Vol. 41, 1982, pp. 383-401. Zbl0484.49012MR601890
- [23] H.J. Sussmann, A Cornucopia of Abnormal Sub Riemannian Minimizers. Part I: the Four-Dimensional Case, IMA Preprint Series #1073,1992. MR1184651
- [24] H.J. Sussmann and W. Liu, Shortest Paths for Sub-Riemannian Metrics on Rank-2 Distributions, Preprint, 1993.
- [25] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Chelsea, New York, 1980.
- [26] A.A. Agrachev and A.V. Sarychev, Strong Minimality of Abnormal Geodesics for 2-Distributions, J. of Dynamical and Control Systems, Vol. 1, 1995, pp. 139-176. Zbl0951.53029MR1333769
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